I have defined a lift of a function $f: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$ as $\hat{f}(t)=f_0+\int_0^t f'([s]) ds$, where $f_0$ is the lift to $\mathbb{R}$ of $f([0])$ and $f'\in\mathbb{R}$ is well defined taking the same lift in the definition $$ f'(\theta)=\lim_{\epsilon \rightarrow 0} \frac{\hat{f}(\theta+\epsilon)-\hat{f}(\theta)}{\epsilon}. $$ Any idea of how to generalize this definition of the lift and derivative to the case $f:\mathbb{R}^n/\mathbb{Z}^n \rightarrow \mathbb{R}^n/\mathbb{Z}^n$? Thank you!
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How does your lift differ from just setting $\hat f(t) = f([t])$? Can you give an example where they'll differ? – md2perpe Jun 01 '17 at 16:32
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As an answer to your question, if $t = (t_1, \ldots, t_n)$ set $\hat f(t) = (\hat f(t_1), \ldots, \hat f(t_n))$. – md2perpe Jun 01 '17 at 16:34